I play a D&D 5e campaign. My Tempest Cleric is about to take his first multi-class dip into Wizard.

Am I able to use my Cleric class and domain features on my Wizard spells, and vice-versa? Both at the same time?

Examples:

When I cast Shocking Grasp (Wizard spell), can I use my Channel
Divinity: Destructive Wrath to maximize the lightning damage and then
use Thunderbolt Strike (another domain feature) to cause the target
to be pushed back 10 feet?

When I roll damage on the weapon attack made for Booming Blade
(Wizard spell), can I also add the 1d8 or 2d8 Thunder damage from my
domain’s Divine Strike feature to the total?

If I study the School of Abjuration, is my Arcane Ward replenished if
I cast Freedom of Movement (Cleric abjuration spell, not on Wizard
spell list)?

If I study the School of Evocation, can I cast Shatter (Cleric domain
spell) and apply both Sculp Spell (Wizard School of Evocation
feature) to avoid hitting my allies in the blast radius and ALSO use
Channel Divinity: Destructive Wrath to maximize the thunder damage?

Sculpt Spell and Channel Divinity: Destructive Wrath on shatter

When you cast an evocation spell that affects other creatures that you can see [...]

Shatter is an evocation spell, so you can use Sculpt Spell; it deals thunder damage, so you can use Destructive Wrath. This works.

Examples that don't work

Light Domain cleric's 8th level ability, Potent Spellcasting states:

... you add your Wisdom modifier to the damage you deal with any cleric cantrip.

As this ability specifically calls out that the cantrip must be a cleric spell it will not work with your multiclassed cantrips.

Thanks to MivaScott for the example

Other considerations

Unlike previous editions, 5th edition makes no distinction between the sources of your spell slots for multiclassed spellcasters, with the slight exception of warlocks as Pact Magic works differently. However, even for warlocks, you can cast any prepared/known spell from any class in any slot of the appropriate level.

\$\begingroup\$As an example of where spells from other classes don't mix, a Light Cleric gains "Potent Spellcasting" at level 8. It specifically calls out, "...you add your Wisdom modifier to the damage you deal with any cleric cantrip." Which is odd because there is exactly one damaging Cleric cantrip in the PHB. But if you cast the Wizards's Fire Bolt cantrip it would not gain the bonus.\$\endgroup\$
– MivaScottJan 17 at 17:11

\$\begingroup\$@MivaScott thankyou, I knew there were counter examples but I hadn't had time to find one.\$\endgroup\$
– linksassinJan 17 at 22:27

\$\begingroup\$Another example that specifies the class (not to turn this into an example farm) is the Wild Magic Sorcerer's Wild Magic Surge feature, which only triggers when casting (leveled) sorcerer spells. And Evocation Wizards' Empowered Evocation and Overchannel features only apply to wizard spells.\$\endgroup\$
– V2BlastJan 17 at 23:33